Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.Comment: 40 pages. Corrected typos and improved redactio

Myocardial energy metabolism and ultrastructure with polarizing and depolarizing cardioplegia in a porcine model

OBJECTIVES: This study investigated whether the novel St. Thomas’ Hospital polarizing cardioplegic solution (STH-POL) with esmolol/adenosine/magnesium offers improved myocardial protection by reducing demands for high-energy phosphates during cardiac arrest compared to the depolarizing St. Thomas’ Hospital cardioplegic solution No 2 (STH-2). METHODS: Twenty anaesthetised pigs on tepid cardiopulmonary bypass were randomized to cardiac arrest for 60 min with antegrade freshly mixed, repeated, cold, oxygenated STH-POL or STH-2 blood cardioplegia every 20 min. Haemodynamic variables were continuously recorded. Left ventricular biopsies, snap-frozen in liquid nitrogen or fixed in glutaraldehyde, were obtained at Baseline, 58 min after cross-clamp and 20 and 180 min after weaning from bypass. Adenine nucleotides were evaluated by high-performance liquid chromatography, myocardial ultrastructure with morphometry. RESULTS: With STH-POL myocardial creatine phosphate was increased compared to STH-2 at 58 min of cross-clamp [59.9 ± 6.4 (SEM) vs 44.5 ± 7.4 nmol/mg protein; P < 0.025], and at 20 min after reperfusion (61.0 ± 6.7 vs 49.0 ± 5.5 nmol/mg protein; P < 0.05), ATP levels were increased at 20 min of reperfusion with STH-POL (35.4 ± 1.1 vs 32.4 ± 1.2 nmol/mg protein; P < 0.05). Mitochondrial surface-to-volume ratio was decreased with polarizing compared to depolarizing cardioplegia 20 min after reperfusion (6.74 ± 0.14 vs 7.46 ± 0.13 µm2/µm3; P = 0.047). None of these differences were present at 180 min of reperfusion. From 150 min of reperfusion and onwards, cardiac index was increased with STH-POL; 4.8 ± 0.2 compared to 4.0 ± 0.2 l/min/m2 (P = 0.011) for STH-2 at 180 min. CONCLUSIONS: Polarizing STH-POL cardioplegia improved energy status compared to standard STH-2 depolarizing blood cardioplegia during cardioplegic arrest and early after reperfusion.publishedVersio

The rolling problem: overview and challenges

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the development of intrinsic Riemannian geometry, from geometric control theory and sub-Riemannian geometry. We also mention how other areas -such as robotics and interpolation theory- have employed the rolling model.Comment: 20 page